A045630 - OEIS

A045630

Number of 2n-bead black-white complementable strings with n black beads and fundamental period 2n.

1, 1, 2, 9, 32, 125, 450, 1715, 6400, 24300, 92250, 352715, 1351584, 5200299, 20056582, 77558625, 300533760, 1166803109, 4537542888, 17672631899, 68923172000, 269128935495, 1052049129142, 4116715363799, 16123800483072, 63205303218750, 247959261273750, 973469712799746

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OFFSET

0,3

COMMENTS

For n>0, a(n) is divisible by n (cf. A022553), 2*a(n) is divisible by n^2 (cf. A268619), and 12*a(n) is divisible by n^3 (cf. A268592). - Max Alekseyev, Feb 07 2016

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..1664

Ashleigh Adams and Esther Banaian, The Cyclic Sieving Phenomenon and frieze patterns, arXiv:2509.17258 [math.CO], 2025. See p. 20.

FORMULA

Moebius transform of A001700(n-1). - Christian G. Bower.

For n>0, a(n) = A007727(n)/2 = n * A022553(n).

MATHEMATICA

a[n_] := 1/2*DivisorSum[n, MoebiusMu[n/#]*Binomial[2 #, #] &]; a[0] = 1; Array[a, 28, 0] (* Michael De Vlieger, Sep 26 2025 *)

CROSSREFS

Cf. A007727, A022553.

Sequence in context: A076959 A003697 A006253 * A075364 A026526 A086607

Adjacent sequences: A045627 A045628 A045629 * A045631 A045632 A045633